Topological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of Topological Machine Learning, great attention has been devoted to signal processing of such topological signals. Most of the previous topological signal processing algorithms treat node and edge signals separately and work under the hypothesis that the true signal is smooth and/or well approximated by a harmonic eigenvector of the Hodge-Laplacian, which may be violated in practice. Here we propose Dirac-equation signal processing, a framework for efficiently reconstructing true signals on nodes and edges, also if they are not smooth or harmonic, by processing them jointly. The proposed physics-inspired algorithm is based on the spectral properties of the topological Dirac operator. It leverages the mathematical structure of the topological Dirac equation to boost the performance of the signal processing algorithm. We discuss how the relativistic dispersion relation obeyed by the topological Dirac equation can be used to assess the quality of the signal reconstruction. Finally, we demonstrate the improved performance of the algorithm with respect to previous algorithms. Specifically, we show that Dirac-equation signal processing can also be used efficiently if the true signal is a non-trivial linear combination of more than one eigenstate of the Dirac equation, as it generally occurs for real signals.

翻译：拓扑信号是与网络节点和边相关联的变量或特征。近年来，在拓扑机器学习的背景下，此类拓扑信号的信号处理受到了极大关注。以往大多数拓扑信号处理算法将节点信号与边信号分开处理，并基于真实信号具有平滑性和/或可由霍奇-拉普拉斯算子的调和特征向量良好逼近的假设，而这一假设在实践中可能并不成立。本文提出狄拉克方程信号处理框架，通过联合处理节点与边信号，能够高效重构真实信号——即使这些信号既不平滑也不调和。该受物理学启发的算法基于拓扑狄拉克算子的谱特性，利用拓扑狄拉克方程的数学结构来提升信号处理算法的性能。我们讨论了拓扑狄拉克方程遵循的相对论性色散关系如何用于评估信号重构的质量。最后，我们通过实验证明该算法相较于以往算法的性能提升。具体而言，我们展示了当真实信号是狄拉克方程多个本征态的非平凡线性组合时（真实信号通常满足此条件），狄拉克方程信号处理仍能保持高效处理能力。